To investigate the sensing characteristics aspects associated with the designed sensor, absorption spectra were modeled and optimized using a commercial electromagnetic solver (CST Microwave Studio). The structural cells are set up as periodic boundaries for X and Y directions and opening space boundaries for Z direction in the simulation setup to build the cell model, and adaptive meshing is used to mesh the structure to improve the structural simulation accuracy. Since the metal reflective layer with the sensor thickness of 0.5 µm can effectively block the propagation of terahertz waves, the transmittance of the sensor to terahertz electromagnetic waves T(ω) = 0. The absorption rate of the metamaterial absorber can be expressed by the following equation.

*A*(ω) = 1-*R*(*ω*)-*T*(*ω*) = 1-|*S*11|2- |*S*21|2=1-|*S*11|2 (1)

Among them, R(ω) and T(ω) denote the reflectance and transmittance of the metamaterial absorber, respectively, and S11 and S21 denote the reflectance and transmittance coefficients of the metamaterial absorber, respectively. Figure 3 shows the S11 and absorbance curves of the sensor in TE mode and TE mode with forwarding electromagnetic wave incidence, as can be seen from the graph, the sensor forms an absorption peak of up to 98.9% in the 0.3–0.6 THz band and the S11 and absorbance curves in both modes overlap perfectly.

## 3.1 Analysis of the absorption mechanism

From Fig. 3, the sensor is shown to form 98.9% of the absorption peak over a resonant frequency of 0.4696 THz. The physical mechanism of the peak absorption of the sensor allows the analysis of the distribution of surface electric currents and energy losses at the resonant frequency point to be explored. The distribution of surface currents and energy loss in the X-Y plane of the sensor under the vertical incidence of terahertz electromagnetic waves in TE and TM modes is shown in Fig. 4. As can be seen in Figs. 4(a) and 4(e), the four open square rings of the metal structure of the sensor produce induced currents in the TE mode for a resonant frequency of 0.4696 THz, with the left and right open square rings producing induced currents in opposite directions, significantly stronger than the top and bottom metal square rings; in TM mode, the induced currents generated by the upper and lower open square rings are in opposite directions and significantly stronger than In TM mode, the induced currents generated by the upper and lower open square rings are in opposite directions, and the intensity is significantly stronger than that of the two open square rings. As can be seen from Figs. 4(b) and 4(f), in TE mode and TM mode, the direction of the induced current on the front of the metal structure is opposite to that on the metal back plate, and the two opposite surface currents form a loop. The direction of the magnetic field excited by this current ring coincides with that of the magnetic polarisation of terahertz electromagnetic waves, and the magnetic dipole effect between the two occurs, forming a magnetic resonance. When the induced current is generated on the sensor surface, it is known from the power loss equation Ploss = I2R (where Ploss shows the electro-magnetic power loss, I is the current at the surface and R is the electrical resistance at the surface) that the induced current on the sensor surface will consume the electromagnetic wave in the form of thermal dissipation due to ohmic loss. From Figs. 4(c) and 4(g), it can be seen that the electromagnetic wave loss in TE mode is mainly caused by the two open square rings on the right and left of the metal structural layer; the electromagnetic wave loss in TM mode is mainly caused by the two open square rings on the top and bottom of the metal structural layer.

To further analyze the absorption mechanism for the transducers, the equivalent impedance of the transducers was analyzed utilizing the impedance matching principle. When a terahertz electromagnetic wave is projected onto the sensory sensor, part of the incident electromagnetic wave is reflected off the surface of the sensor and another part is transmitted through. It follows that to achieve maximum absorption, the reflection and transmission coefficients must be minimized. In general, sensors usually use a full metal backing plate of much greater thickness than the skin depth as their ground plane, so that transmission is near zero. Thus, the absorptance of the sensor is determined mainly by the amount of reflectance.

The equivalent permittivity and equivalent permeability of the metamaterial absorber can be expressed as:

*ε*(*ω*) = *ε*'-*jε*" (1)

*µ*(*ω*) = *µ*'-*µ*" (2)

Among them, ε' and µ' are the parameters describing the degree of polarization and magnetization, respectively, and ε" and µ" denote the electric and magnetic losses of the metamaterial absorber, respectively.

When the dimensions of the resonant structural element of the sensor are smaller than the wavelength at the corresponding operational frequency, the sensor as a whole can be considered as an equivalent medium, and therefore the equivalent impedance of the sensor can be derived from the equivalent capacitive weight µ (ω) and the equivalent magnetic permeability ε (ω). Can be expressed as:

$$Z\left( \omega \right)={\text{ }}\sqrt {\mu \left( \omega \right)/\varepsilon (\omega )}$$

3

The relationship between the reflection coefficient S11, transmission coefficient S21, and equivalent impedance of the sensor can be expressed as:

$$Z\left( \omega \right){\text{=}}\sqrt {\frac{{{{\left( {{\text{1+}}{S_{11}}} \right)}^2} - {S_{21}}^{2}}}{{{{\left( {{\text{1-}}{S_{11}}} \right)}^2} - {S_{21}}^{2}}}}$$

4

When the electromagnetic wave is incident on the sensor surface, the degree of matching between its equivalent impedance and the free space impedance determines the magnitude of the reflection coefficient, so the reflection coefficient R(ω) can be expressed as:

$$R\left( \omega \right)={\text{ }}{\left| {{S_{11}}} \right|^2}={\text{ }}{\left( {\frac{{Z\left( \omega \right) - 1}}{{Z\left( \omega \right)+1}}} \right)^2}={\text{ }}\frac{{{{\left[ {(\operatorname{Re} \left\{ {Z\left( \omega \right)} \right\} - {Z_0}\cos \theta )} \right]}^2}+{{\left[ {\operatorname{Im} \left\{ {Z\left( \omega \right)} \right\}} \right]}^2}}}{{{{\left[ {(\operatorname{Re} \left\{ {Z\left( \omega \right)} \right\}+{Z_0}\cos \theta )} \right]}^2}+{{\left[ {\operatorname{Im} \left\{ {Z\left( \omega \right)} \right\}} \right]}^2}}}$$

5

where Z0 is the wave impedance of the incident wave in free space, Z0 ≈ 377 Ω, and θ is the angle of incidence of the electromagnetic wave. Thus, for a vertically incident electromagnetic wave with an incidence angle θ of 0, the reflectivity will be near zero as the normalized equivalent compound of the absorber impedance approaches 1.

The reflection coefficient R(ω) is obtained by rectifying the formula:

$$R\left( \omega \right)=\frac{{{Z_L} - {Z_0}}}{{{Z_L}+{Z_0}}}$$

6

Among them, ZL is the equivalent impedance of the sensor, when ZL = Z0 to meet the impedance matching, the reflection coefficient R(ω) = 0 for the electromagnetic wave irradiated to the sensor in space. It is when the total impedance of a sensor is closer to the free-standing impedance of air that most of the electromagnetic energy is guaranteed to enter the sensor. The equivalent complex impedance frequency spectrum of the sensor is shown in Fig. 5. The equivalent complex impedance of the sensor at a resonant frequency of 0.4695 THz is ZL = 1.1 + i*0.018, while the complex free-space impedance is Z0 = 1. As a result, a good match of impedance is formed at the resonant frequency between the sensor on the one hand and the free space on the other, resulting in a high absorption resonance peak.

## 3.2 Incidence angle and polarization-sensitive characteristics analysis

In practical applications, sensors with wide incidence angle and polarization insensitive characteristics can improve detection accuracy and reduce experimental errors, thus improving detection efficiency. Designing the sensing unit into a quadruple rotationally symmetric structure can realize the polarization-insensitive characteristics of the sensor. Figures 6(a) and (b) show the effect of the variation of the incident angle of the terahertz wave on the absorption rate of the sensor in the TM and TE modes, respectively. In TE mode, the absorptance of the sensor at the resonant frequency 0.4695 THz is higher than 96% for all the terahertz wave incident angles in the range of 0–30°, and no significant shift of the resonant frequency occurs. In the TM mode, the absorption rate at the resonant frequency 0.4695 THz was higher than 90% for both terahertz wave incident angles in the range of 0–30°, and no significant shift in the resonant frequency occurred. In addition, the sensor produced additional resonant peaks with increasing resonance rate as the angle of incidence of terahertz waves increased in TE and TM modes. This is because the resonance generated at the sensor surface is increasing with the increasing incident angle, resulting in additional resonance peaks. Since the additional resonance peak is far from the resonance frequency of 0.4695 THz, it does not interfere with the sensor detection. The above-mentioned results showed the sensor maintains a high absorption rate when terahertz waves are incident at angles from 0 to 30° and has a wide incidence angle insensitivity. Figures 6(c) and (d) give the effect of the variation of polarization angle on the absorptance of the sensor in TE and TM modes. The absorptance and resonant frequency of the sensor remain unchanged under the vertical incidence of terahertz waves with a 0–90° polarization angle, so the sensor has good polarization insensitivity characteristics.

## 3.3 Sensing performance analysis

The operating principle of the designed sensor is that the injection of liquid phase analyses possessing different refractive indices into a sensor will result in a change in the dielectric parameters of the sensor, which in turn will change the resonant characteristics of the sensor (resonant frequency, absorbance, resonance peak, etc.).

Q, S, and FOM values represent the three indicators of sensor performance. The Q value indicates a sensor's resonance characteristics, the size of which is relevant to both resolutions the sensitivity. In general, the higher the Q value, the smaller the dielectric loss of the structure and the narrower and sharper the resonance peak. The FOM value measures the overall performance of the sensor, and the higher the FOM value, the better the sensor performance. The formula for its calculation is as follows.

Here, f denotes the central resonant frequency and FWHM (full-width half height) denotes the half-peak width.

$$S=\frac{{\Delta f}}{{\Delta n}}$$

8

Here, Δn represents the amount of index of refraction change of the analyte, Δf represents the magnitude of the frequency shifts of the centroid frequency and the sensitivity is in GHz/RIU (refractive index units).

$$FOM=\frac{S}{{FWHM}}$$

9

Liquid analytes with different refractive indices were injected into the sensor and a series of simulations were performed to investigate their absorption characteristics. The refractive indices of the analytes were set to 1.0 to 2.0 as many biological macromolecules range from 1.0 to 2.0. The simulation results in a redshift with increasing refractive power of the analytical material and the resonance peak of the sensor is almost always above 99% absorption, as shown in Fig. 7(a). Figure 7(b) shows the results of a linear fit to the resonant peak of the sensor with a sensitivity of S = 78.6 GHz/RIU; Figs. 7(c) with 7(d) show the influence of the refractive index of the analyte upon the sensor's Q and FOM values and it can be seen that as the refractive index increases, the Q of the resonant peak fluctuates slightly above and below 55 and the FOM value fluctuates slightly above and below 9.6. When the refractive index of the analyte is 1.4, the Q value of the sensor is 55.32 and the maximum value of the FOM is 9.81. The excellent resonance characteristics indicate that the designed sensor has excellent performance and can achieve the identification of biomolecules with different refractive indices.

The terahertz sensor can shorten the detection time and improve the detection accuracy when detecting analytes in solution, enabling fast and accurate detection of liquid-phase analytes. The effect of the tangent to the loss angle of the analyte on the absorptance of the sensor when the real part of the analyte's dielectric constant is 1 is shown in Fig. 8. It is seen that as the loss angle of the analyzed object increases tangentially, the resonant frequency of the sensor has a small redshift, and the absorbance and resonant intensity decrease simultaneously. The FWHM and Q values of the absorption peak also decrease with the increase of the loss angle tangent. Therefore, the sensor can distinguish the change in loss angle of the object to be measured based on the change in absorbance and Q-value. Table 2 shows a comparison of the sensitivity, Q value, and FOM values of the proposed sensor and terahertz sensors designed in the literature. It can be seen that the designed sensor has high Q and FOM values for analytes with different refractive indices, which indicates that the sensor has good accuracy and stability and has good prospects for applications in bio-detection and medical prevention.

Table 2

Comparison of sensor sensitivity, Q value, and FOM value between the sensor proposed in this paper and those in the references

References | Operating band/THz | Sensitivity/(GHz/RIU) | Quality factor | Figure of merit |

29 | 1 ~ 2.2 | 300 | 22.05 | 2.94 |

30 | 5 ~ 6.5 | 1800 | 5.92 | 15 |

31 | 0.8 ~ 1.8 | 243 | 14.2 | 3.3 |

32 | 0.2 ~ 1 | 124 | 6.913 | Not given |

33 | 0.4 ~ 1.2 | 76.5 | Not given | Not given |

34 | 0.2 ~ 1 | 74 | Not given | Not given |

Proposed | 0.3 ~ 0.6 | 78.6 | 55.3 | 9.81 |